#8395258

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#49994	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,36)(27,33,28,32)(29,38,39,34)(30,35,40,37);(1,2)(3,8)(4,7)(5,6)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,27,31,28)(26,32,36,33)(29,35,39,37)(30,34,40,38);(2,6)(4,8)(12,20)(15,22)(18,24)(21,23)(25,27,31,28)(26,32,36,33)(29,37,39,35)(30,38,40,34);(2,6)(3,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28,31,27)(26,33,36,32)(29,37,39,35)(30,38,40,34);(1,3,5,7)(2,4,6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,29)(26,34)(27,37)(28,35)(30,32)(31,39)(33,40)(36,38);(1,4)(2,7)(3,6)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,30,31,40)(26,35,36,37)(27,34,28,38)(29,32,39,33)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8302454);